1. “Dark energy” in the Local Void
M. Villata
arXiv:1201.3810v1 [astro-ph.CO] 18 Jan 2012
Abstract The unexpected discovery of the accelerated 1 Introduction
cosmic expansion in 1998 has filled the Universe with
the embarrassing presence of an unidentified “dark en- Since the end of the last century, observations of high-
ergy”, or cosmological constant, devoid of any phys- redshift type Ia supernovae have unexpectedly shown
ical meaning. While this standard cosmology seems that the cosmic expansion is currently in an accel-
to work well at the global level, improved knowledge eration phase (e.g. Riess et al. 1998; Perlmutter et al.
of the kinematics and other properties of our extra- 1999), whose physical cause is unknown. Formally, this
galactic neighborhood indicates the need for a better acceleration is ascribed to an additional term having
theory. We investigate whether the recently suggested a negative pressure in the expansion equations, which
repulsive-gravity scenario can account for some of the represents about 75% of the total energy density of the
features that are unexplained by the standard model. Universe, in the simplest case corresponding to a cos-
Through simple dynamical considerations, we find that mological constant, perhaps associated to the energy
the Local Void could host an amount of antimatter of the quantum vacuum. Besides this standard cos-
(∼ 5×1015 M⊙ ) roughly equivalent to the mass of a typ- mology of the ΛCDM model, various alternatives have
ical supercluster, thus restoring the matter-antimatter been proposed to explain the cosmic speed-up, invok-
symmetry. The antigravity field produced by this “dark ing scalar fields or modifications of general relativity,
repulsor” can explain the anomalous motion of the Lo- such as extensions to extra dimensions or higher-order
cal Sheet away from the Local Void, as well as several curvature terms (e.g. Amendola 2000; Dvali et al. 2000;
other properties of nearby galaxies that seem to require
Carroll et al. 2004; Capozziello et al. 2005). In a vari-
void evacuation and structure formation much faster
ant of these alternative theories, Villata (2011) proposes
than expected from the standard model. At the global
to extend general relativity to antimatter, intended as
cosmological level, gravitational repulsion from anti-
CPT-transformed matter, whose immediate result is
matter hidden in voids can provide more than enough
the prediction of a gravitational repulsion between mat-
potential energy to drive both the cosmic expansion and
ter and antimatter, which, with antimatter hidden in
its acceleration, with no need for an initial “explosion”
cosmic voids, could explain the accelerated expansion.
and dark energy. Moreover, the discrete distribution of
these dark repulsors, in contrast to the uniformly per- Knowledge of peculiar motions and spatial distribu-
meating dark energy, can also explain dark flows and tion of galaxies allows us to explore gravitational inter-
other recently observed excessive inhomogeneities and actions within and among clusters. This is particularly
anisotropies of the Universe. true for our extragalactic neighborhood, where dis-
tances can be measured with higher precision. Through
Keywords Gravitation — Cosmology: theory — Dark a study performed on a database of about 1800 galaxies
energy — Large-scale structure of Universe within 3000 km s−1, Tully et al. (2008) find that the pe-
culiar velocity of the Local Sheet decomposes into three
M. Villata dominant, almost orthogonal components. One of them
INAF, Osservatorio Astronomico di Torino, Via Osservatorio 20, (of 185 km s−1) is ascribed to the gravitational pull of
I-10025 Pino Torinese (TO), Italy the Virgo Cluster and its surroundings, while a second,
e-mail: villata@oato.inaf.it larger component (of 455 km s−1) seems to be due to an
attraction on larger scales nearly in the direction of the

2. 2
Centaurus Cluster. The third component (259 km s−1) Ni 2004; Hajdukovic 2010, 2011c,b; see also Cabbolet
is not directed toward anything prominent, but points 2010), or that it is even self-repulsive (e.g. Noyes 2008;
away from the Local Void. Some evacuation of voids is Benoit-L´vy & Chardin 2012).
e
expected in standard cosmology, due to the density con- In a recent paper, Villata (2011) argued that there
trast that pushes matter outward, but the high speeds is no need to change the sign of the gravitational mass
of galaxies at the edge of the Local Void indicate that it of antimatter (which would represent a violation of the
should be very large and very empty, more than usually weak equivalence principle) to get repulsion between
expected. Consequently, alternative models have been matter and antimatter; but he showed that antigravity
considered. For instance, Dutta & Maor (2007) argue appears as a prediction of general relativity, once it is
that dark energy with a varying equation of state could assumed that this theory is CPT invariant and that,
have higher density where mass density is lower, so that consequently, matter is transformed into antimatter by
repulsion from voids would be stronger. these three joint operations (charge conjugation, parity,
Moreover, as discussed by Peebles & Nusser (2010), and time reversal).
many other properties of galaxies in our neighborhood Besides representing the universal symmetry of field
are not explained by the standard model. Besides the theories, the CPT transformation is consistent with the
observed extreme emptiness of the Local Void (∼ 10−5 Feynman-Stueckelberg interpretation of antiparticles as
probability), the presence of large, luminous galaxies particles traveling backwards in time, since this view
in the adjacent low-density regions is in contrast with explains the need of applying T (and P) in addition
expectations (< 1% probability). These and other ob- to C to describe the behavior of antimatter. In other
served features seem to indicate the need for a better words, a physical system (or a component of a physi-
theory, entailing a mechanism by which matter is more cal system) traveling in the opposite time direction is
rapidly repelled by voids and gathered into galaxies and observed from our time direction through a CPT trans-
structures. formation, and this would be the reason why physical
In this paper we want to investigate whether the re- laws are CPT symmetric, since they must describe the
pulsive gravity scenario proposed by Villata (2011) as a behavior of physical systems as observed from either
potential explanation for the cosmic acceleration could time directions, i.e. when they appear to be composed
also help solve some of these problems. A brief intro- of either matter or antimatter, or both.
duction to antimatter gravity is given in Sect. 2, while In general relativity, the equation of motion for a
in Sect. 3 this view is applied to the dynamics of our matter test particle in a matter-generated gravitational
extragalactic surroundings. Discussion and conclusions field is
can be found in Sect. 4.
d2 xλ dxµ dxν
2
= −Γλ
µν . (1)
dτ dτ dτ
2 Antimatter gravity If we CPT-transform all the four elements in Eq. (1),
we get an identical equation describing the motion of
The existence of antigravity, i.e. gravitational repulsion an antimatter test particle in an antimatter-generated
between matter and antimatter, has been debated at gravity field, since all the four changes of sign cancel one
length during the last several decades, without reach- another. Thus, this CPT symmetry ensures the same
ing any firm conclusion. After strong opposition dur- self-attractive gravitational behavior for both matter
ing the second half of last century, the idea of anti- and antimatter. But, if we transform only one of the
gravity has experienced a period of renewed interest two components, either the field Γλ or the particle
µν
because of the discovery of the accelerated expansion (represented by the remaining three elements), we get
of the Universe in 1998. In particular, against anti- a change of sign that converts the original gravitational
gravity in the Fifties–Sixties were proposed some the- attraction into repulsion, so that matter and antimatter
oretical arguments that seemed to rule out its exis- repel each other.
tence (Morrison 1958; Schiff 1958, 1959; Good 1961), The equation for a massless particle (e.g. a photon)
but which were later criticized and questioned (e.g. is formally equal to Eq. (1), except for the parameter
Nieto & Goldman 1991; Chardin & Rax 1992; Chardin τ , which can no longer be taken as the proper time,
1993, 1997; Hajdukovic 2011a), and then lost much being dτ = 0, but it will be another parameter describ-
of their effectiveness. At the same time, several au- ing the world line. Therefore, a (retarded) photon will
thors have pursued the idea of antigravity, either assum- be repelled by an antimatter gravity field, as well as
ing that antimatter has negative gravitational mass, a CPT-transformed photon, i.e. an advanced photon,
and thus is self-attractive (e.g. Noyes & Starson 1991;

3. 3
will be repelled by matter1 . This may provide a test which can be decomposed into three quasi-orthogonal
for the theory of antigravity: the presence of antimat- components (Tully et al. 2008). One of them is due
ter in cosmic voids suggested by Villata (2011, 2012) to the well-known attraction towards the Virgo Clus-
might be revealed by its gravitational effect on the ra- ter and its dense surroundings (e.g. Tonry & Davis
diation coming from background sources, in a sort of 1981; Aaronson et al. 1982; Hoffman & Salpeter 1982;
antigravitational lensing. Tully & Shaya 1984; Tonry et al. 2000). The sec-
In the Newtonian approximation, the law of gravity ond, largest component has a less clear origin: it is
extended to antimatter becomes F (r) = ∓GmM/r2 , ascribed to gravitational pull on scales larger than
where the minus sign refers to the gravitational self- 3000 km s−1 nearly in the direction of the Centau-
attraction of both matter and antimatter, while the rus Cluster, but the identification and relative impor-
plus sign indicates the gravitational repulsion between tance of the “great attractors” is still debated (e.g.
matter and antimatter. Shaya 1984; Lynden-Bell et al. 1988; Scaramella et al.
The change of sign in the CPT-transformed (i.e. 1989; Raychaudhury 1989; Kocevski & Ebeling 2006;
antimatter-generated) gravitational field comes from Erdo˘du et al. 2006).
g
the inversion of the derivatives of the metric tensor The last component corresponds to what is his-
in the expression of Γλ . Thus, while the metric ten-
µν torically known as the “local velocity anomaly” (e.g.
sor, i.e. the potential, does not change sign and re- Faber & Burstein 1988; Tully 1988; Tully et al. 1992).
mains the same for matter and antimatter, the sign It is large, 259 km s−1, and is anomalous because is not
change appears only in the field. This means that in directed towards any important structure. It can not
the above generalized Newton law, the plus sign indi- be due to a simple gravitational pull, otherwise also the
cating repulsion does not formally come from an in- Leo Spur galaxies, which would be in between, should
verted potential, but from its inverted gradient. In- be affected by it, while they appear to be at rest with
deed, the potential φ is usually defined as −∇φ = g, respect to this motion. Thus, the cause of this compo-
so that, being the attractive field g(r) = −GM/r2 , nent must be on the other side, where the Local Sheet
the potential is φ(r) = −GM/r, and, with the same borders the Local Void. Moreover, this velocity vec-
potential, the repulsive field comes from its inverted tor points just in the opposite direction with respect to
gradient: g (r) = GM/r2 . This formally correct view
¯ an approximate center of the void, and is orthogonal to
is actually not intuitive, so that it is preferable to re- the disk-like structure of the Local Sheet. Whatever the
verse the potential and to get the field according to the cause of the anomalous motion, it must affect the Local
usual −∇φ definition. The repulsive potential is thus Sheet more than the rest of our neighborhood (which
¯
φ(r) = GM/r, and the gravitational picture of the cos- is indeed more distant from the void), and also seems
mic alternance of matter and antimatter suggested in to have the property to squash the closest structures,
Villata (2011) can be represented as an alternation of besides giving them high velocities.
potential wells and hills. Of course, from the antimat- These and several other features (see Sects. 1 and
ter point of view this situation would be seen upside 4), such as the extreme emptiness of the Local Void,
down. do not find a satisfactory explanation within the stan-
dard model, and seem to require the intervention of a
repulsive force. In particular, the flattened shape of
3 Dynamics of the Local Sheet the Local Sheet looks just like the signature of a repul-
sive gravitational field. Indeed, while the effect of tidal
Our peculiar motion with respect to the general ex- forces in an attractive gravity field is to stretch struc-
pansion of the Universe can be inferred by the ob- tures along the radial direction, since the side closer to
served dipole anisotropy in the cosmic microwave back- the field source is more attracted than the side farther
ground (e.g. Fixsen et al. 1996). This motion has well- away, the tidal effect of a repulsive field is to squash
known local contributions, including the orbital veloc- them, the near side being more pushed than the far
ity of the Sun in our Galaxy and the attraction of one: dF/dr = −2GmM/r3 .
the latter towards M31. Once these components are Under some simple assumptions in linear perturba-
taken into account, it is found that the Local Sheet tion theory, the peculiar velocity field Vpec of an ideal
has still a very large peculiar velocity of 631 km s−1, pressureless fluid is related to the gravitational acceler-
ation g and the expansion time t by the simple formula
1 As a consequence, the energy of a retarded-advanced photon
(see e.g. Peebles 1993; Peebles et al. 2011) Vpec = g t.
pair in a gravitational field would be conserved, thus invalidating We can check whether this equation can represent a rea-
the Morrison (1958) argument against antigravity. sonable approximation for the peculiar velocity of the

4. 4
Local Sheet with respect to the gravitational attraction the total energy per unit mass, given by V 2 /2 + GM/d
by the Virgo Cluster. Rewriting it as (where V = H0 d + Vpec is the total velocity with re-
spect to the void center) at the Local Sheet and Leo-
Vpec M Mpc2 t
= 4.4 × 10−12 , (2) Spur/Virgo distances. In the intermediate case seen
km s −1 M⊙ d2 Gyr above, with MLV = 6.6 × 1015 M⊙ , dLS = 25 Mpc
LS
with t = t0 = 13.75 Gyr (Jarosik et al. 2011), d = and Vpec = 634 km s−1 for the Local Sheet, dLeo =
17 Mpc and Vpec = 185 km s−1 (Tully et al. 2008), we 32.5 Mpc and Vpec = 375 km s−1 for Leo-Spur/Virgo2,
Leo
get for the mass of the Virgo Cluster and surroundings and adopting H0 = 74 km s−1 Mpc−1 from Tully et al.
MVir = 8.8 × 1014 M⊙ . This value is in good agree- (2008), we obtain 4.2×106 and 4.7×106 km2 s−2 for the
ment with previous estimates (e.g. Tully & Shaya 1984; respective total specific energies. We then find a small
Mohayaee & Tully 2005), so that we may be confident difference of about 10%, which could also be due to
in Eq. (2), at least at the scale of our supercluster. the various uncertainties affecting both measures and
Assuming that Eq. (2) holds also for repulsive accel- model assumptions, or could come from slightly differ-
eration, we can try a rough estimate of the mass of anti- ent initial conditions for the differently located systems.
matter (see Sect. 2), possibly located around the center This approximate energy conservation at different dis-
of the Local Void, needed to push the Local Sheet at tances from the antigravity center confirms that the
the observed velocity of 259 km s−1. The expected dis- dynamics in the radial direction is actually dominated
tance of the Local Sheet from the center of the void may by this repulsive interaction, and supports our scenario.
be assumed in the range 25 ± 5 Mpc (e.g. Tully et al. If we go back in time conserving mechanical energy,
2008), which gives MLV = 2.7+1.2 × 1015 M⊙ . How-
−1.0 we find V = 0 at d = 6.7 Mpc for the Local Sheet, and
ever, it would be strange that this repulsive interaction at d = 6.0 Mpc for Leo Spur, whereas one could expect
affected only the Local Sheet, leaving the rest of our much smaller distances, approaching a singularity. This
neighborhood unaffected and at rest with respect to might mean that our data are too imprecise to allow a
the Local Void. Indeed, what is measured as a peculiar similar extrapolation in the past, or that our simplistic
velocity of 259 km s−1 is actually the peculiar velocity dynamical assumption of a dominant, point-like-source
of the Local Sheet relatively to the rest of the Local field can no longer hold at these earlier stages and also
Supercluster, i.e. mainly to the Virgo Cluster, and in surrounding fields should be taken into account, or that
particular to the Leo Spur galaxies, which are both far- some of the energy has been somehow dissipated, or a
ther than the Local Sheet from the approximate cen- combination of all these things.
ter of the Local Void by about 7–8 Mpc. Thus, for In any case, what we are finding throughout this sec-
a more significant estimate of the mass in the Local tion is that the repulsive gravity field can account not
Void, we take the difference of Eq. (2) between the dis- only for peculiar (anomalous) velocities and accelera-
tance of the Local Sheet (again assumed as 25 ± 5 Mpc) tion of the Hubble flow, but even for the Hubble flow
and the Leo-Spur/Virgo radial distance from the void itself. There seems to be more than enough energy for
center (correspondingly estimated as 32.5 ± 4.5 Mpc), driving all the Universe expansion, stored in the poten-
which yields MLV = 6.6+4.6 × 1015 M⊙ , i.e. significantly
−3.1 tial energy, when, going back in time, matter get closer
larger than the former result, but of the same order of and closer to the peaks of the antimatter potential hills.
magnitude. These estimates come from the simplistic No need for initial (artificial) velocities provided by an
assumption that the various components are affected, explosive “Big Bang”, whose energy would not have
in the radial direction from the void center, only by the any identified origin. No need for unknown dark en-
dominant repulsive interaction with an antimatter mass ergy to drive the cosmic acceleration, or dark energy is
concentrated close to the assumed void center, and dis- nothing else than the potential energy of gravitational
regarding other possible contributions in the same di- repulsion.
rection. Thus, we are not claiming that these values
must be considered as extremely realistic, and we can
not even choose between the two results, but we can
4 Discussion and conclusions
say that an antimatter mass of ∼ 5 × 1015 M⊙ , within
a factor of 2, can easily account for the local velocity
There remains the question of why antimatter in voids
anomaly. This value is comfortably comparable to the
should not be visible. It seems that something “dark”
mass of a medium-size supercluster, which means that
must necessarily exist: dark matter, dark energy, and
we would have found the location for antimatter in a
matter-antimatter symmetric Universe.
2 V LS Leo
Another argument in favor of the above picture and pecand Vpec are calculated from Eq. (2), and have the correct
mass estimate is the following one. We can calculate difference of 259 km s−1 between them.

5. 5
now “dark repulsors”. On the other hand, were it oth- of the expanding Universe4 , but at the cost of intro-
erwise, we would have understood everything a long ducing ad hoc elements that do not have a physical
time ago. explanation, like the initial “explosion” causing the cos-
Like matter, antimatter is self-attractive, so we can mic expansion and the dark energy that should accel-
expect that it forms anti-galaxies and anti-stars, which erate it. As we have seen, antigravity can explain both
would emit electromagnetic radiation, we should can the expansion and its acceleration, and this theory also
detect. However, as pointed out by Villata (2012, has the virtue of solving the long-standing problem
see also Sect. 2), antimatter, if emitting, should emit of matter-antimatter asymmetry. It might seem that
advanced radiation, which can be undetectable for at one is replacing something invisible (dark energy) with
least two reasons. First, it could be intrinsically unde- something else equally invisible (dark repulsors). The
tectable, at least with the usual detectors and observing difference is that the presence of the latter is theoreti-
procedures3 . Then, even if it were somehow detectable, cally expected if not required, whilst the presence of the
we know from Sect. 2 that advanced photons are re- former is not requested, unexpected, and embarrassing.
pelled by matter, so that the chance of detecting the Moreover, some recent observational findings are un-
few that possibly succeeded in reaching us on top of the dermining the standard model (e.g. Walker & Pe˜ arrubia
n
potential hill of our Galaxy is further greatly reduced. 2011 on unexpectedly uniform distribution of dark mat-
The possibility remains that antimatter emits also re- ter in dwarf galaxies5 ). In particular, the existence
tarded radiation (as well as matter might emit advanced of “dark flows”, i.e. large-scale, peculiar bulk motions
radiation). In this case we should see antimatter stars, of galaxy clusters (e.g. Kashlinsky et al. 2010 and ref-
galaxies and clusters filling the voids, but well sepa- erences therein), of excess clustering on large scales
rated from matter ones by gravitational repulsion. Yet, (Thomas et al. 2011), of anisotropy in the expansion
as already said, antimatter would be matter traveling acceleration (e.g. Cai & Tuo 2012) seem to indicate the
backwards in time, so that, taking this concept to the need for a distribution of acceleration sources less ho-
extreme, we would observe its cosmic evolution in re- mogeneous than the uniformly permeating dark energy,
verse, and its current evolutionary stage could be very thus favoring the discrete distribution in cosmic voids.
different from ours. For example, if antimatter were Let us return to our extragalactic neighborhood and
“created” in a distant future, now it might be collapsed its anomalies partly mentioned in the Introduction and
into supermassive black holes, or be in some other non- described in detail in Peebles & Nusser (2010). All
emitting form, before definitively collapsing, 14 billion those observed properties, such as the extreme empti-
years ago. Perhaps to be “reborn” as matter, and re- ness of the Local Void, the unexpected presence of large
trace in the opposite direction the time already passed galaxies at its edge, the existence of pure disk galaxies,
through, but well separated from the “anti-itself” by and the insensitivity of galaxy properties to environ-
gravitational repulsion. A further step would lead us ments, indicate the existence of a mechanism that can
to suggest a Universe “recurring over time”, in a cos- produce a fast evacuation of voids and rapid assembly
mic loop, but such a speculative rumination is beyond of matter into structures, which the standard theory
the scope of this paper. based on attractive-only gravity can not provide. Re-
In any case, there seem to be more reasons for anti- pulsive gravity can be the cause of fast evacuation and,
matter invisibility than for visibility, so that we are not also through the squashing tidal effect, can help a lot
surprised not to see anything in cosmic voids. Another to rapidly gather matter into galaxies and structures,
point is the different, quasi-spherical symmetry of voids especially at the borders of voids.
with respect to the squashed/filamentary shape of mat- In conclusion, repulsive gravity between matter and
ter structures. This would support the view of different antimatter, which is a natural outcome of general rela-
evolutionary stages, where antimatter black holes close tivity extended to CPT-transformed systems, is an ex-
to the void centers produce spherical fields, and the cellent candidate to explain not only the cosmic expan-
more uniformly distributed matter can only adapt in sion and its acceleration, but also the anomalous kine-
the potential valleys among the quasi-circular potential matics and several other properties of our extragalactic
hills. surroundings, as well as other observed features of the
Passing several cosmological tests, the ΛCDM stan-
dard model seems to provide a fairly good description 4 However,in a recent paper, Benoit-L´vy & Chardin (2012) have
e
shown that most of these observational constraints can also be
3 Actually there is some confusion about advanced radiation in the explained by their quite different cosmology, based on the as-
literature; for discussions around the issue of advanced radiation sumption of a matter-antimatter symmetric Universe.
in general, see e.g. Davies (1975), Cramer (1980), and references 5 See
Hajdukovic (2011c) for an alternative interpretation of dark
therein. matter based on antigravity.

6. 6
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Perlmutter, S., Aldering, G., Goldhaber, G., et al. 1999,
Astrophys. J., 517, 565
Raychaudhury, S. 1989, Nature, 342, 251
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A